372 SURFACE WAVES. [CHAP. IX 



217. Let us apply this to the free oscillations of a sheet of 

 water, or a straight canal, of uniform depth h, and let us suppose 

 for the present that there are no limits to the fluid in the direction 

 of x, the fixed boundaries, if any, being vertical planes parallel to xy. 



Since the conditions are uniform in respect to #, the simplest 

 supposition we can make is that <f> is a simple-harmonic function 

 of x ; the most general case consistent with the above assumptions 

 can be derived from this by superposition, in virtue of Fourier's 

 Theorem. 



We assume then 



(j) = Pcoskx.e i ^ t+ ^ ........................ (1), 



where P is a function of y only. The equation (1) of Art. 216 

 gives 



whence P = A&+Ber*y ........................ (3). 



The condition of no vertical motion at the bottom is dcf>/dy = 

 for y h, whence 



=, say. 

 This leads to 



$ = C cosh k(y + h] cos kx . e i(<Tt+e) ............... (4). 



The value of a is then determined by Art. 216 (8), which gives 



o- 2 = gk tanh kh ........................ (5). 



Substituting from (4) in Art. 216 (5), we find 



ia-C 

 77 = - cosh kh cos kx.e*^^ ............... (6), 



9 



or, writing a = <rC/g . cosh kh t 



and retaining only the real part of the expression, 



7j = a cos kx . sin (<rt + e) .................. (7). 



This represents a system of ' standing waves,' of wave-length 

 X = 2-7T/&, and vertical amplitude a. The relation between the 

 period (27T/0-) and the wave-length is given by (5). Some numeri- 

 cal examples of this dependence will be given in Art. 218. 



